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Properties

Label 2-140-35.17-c1-0-3
Degree $2$
Conductor $140$
Sign $0.937 + 0.347i$
Analytic cond. $1.11790$
Root an. cond. $1.05731$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.42 − 0.649i)3-s + (−2.19 − 0.407i)5-s + (2.59 − 0.513i)7-s + (2.86 − 1.65i)9-s + (−1.86 + 3.22i)11-s + (−2.90 − 2.90i)13-s + (−5.59 + 0.440i)15-s + (1.78 + 6.67i)17-s + (−1.65 − 2.86i)19-s + (5.96 − 2.93i)21-s + (−1.84 − 0.493i)23-s + (4.66 + 1.79i)25-s + (0.539 − 0.539i)27-s + 0.563i·29-s + (−3.20 − 1.85i)31-s + ⋯
L(s)  = 1  + (1.40 − 0.375i)3-s + (−0.983 − 0.182i)5-s + (0.980 − 0.194i)7-s + (0.953 − 0.550i)9-s + (−0.561 + 0.973i)11-s + (−0.805 − 0.805i)13-s + (−1.44 + 0.113i)15-s + (0.433 + 1.61i)17-s + (−0.380 − 0.658i)19-s + (1.30 − 0.639i)21-s + (−0.383 − 0.102i)23-s + (0.933 + 0.358i)25-s + (0.103 − 0.103i)27-s + 0.104i·29-s + (−0.575 − 0.332i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.937 + 0.347i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ 0.937 + 0.347i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43719 - 0.257589i\)
\(L(\frac12)\) \(\approx\) \(1.43719 - 0.257589i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (2.19 + 0.407i)T \)
7 \( 1 + (-2.59 + 0.513i)T \)
good3 \( 1 + (-2.42 + 0.649i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (1.86 - 3.22i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.90 + 2.90i)T + 13iT^{2} \)
17 \( 1 + (-1.78 - 6.67i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.65 + 2.86i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.84 + 0.493i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 0.563iT - 29T^{2} \)
31 \( 1 + (3.20 + 1.85i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.268 - 1.00i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 7.88iT - 41T^{2} \)
43 \( 1 + (7.53 - 7.53i)T - 43iT^{2} \)
47 \( 1 + (0.543 + 0.145i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.479 - 1.78i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-6.38 + 11.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.00 + 4.62i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.56 + 1.49i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 0.683T + 71T^{2} \)
73 \( 1 + (-11.6 + 3.11i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.76 - 1.59i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.44 - 1.44i)T + 83iT^{2} \)
89 \( 1 + (1.26 + 2.18i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.99 + 3.99i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.96312699425266331296191479954, −12.45397416880384467683095050610, −11.06033574342501929331429647737, −9.924690202094288542329729512495, −8.455420453076110729911115856432, −7.987984768867928761971245030654, −7.20193757551759889954518603893, −4.96211714679441695394713141878, −3.69389968063707294903092063237, −2.11132446299984831838171877217, 2.55913576359065439960679677953, 3.80797961720774962964075506463, 5.07455715191997209020404416082, 7.25225254699946614366269385275, 8.096803319286055041952472892157, 8.795505412230236267219094414675, 9.981143598106385872192414242874, 11.29788720325233679968144182793, 12.03698273968972753784326740982, 13.59188355311599342401139328592

Graph of the $Z$-function along the critical line